Fullerenes with distant pentagons

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1 Fullerenes with distant pentagons Jan Goedgebeur a, Brendan D. McKay b arxiv: v1 [math.co] 12 Aug 2015 a Department of Applied Mathematics, Computer Science & Statistics Ghent University Krijgslaan 281-S9, 9000 Ghent, Belgium jan.goedgebeur@ugent.be b Research School of Computer Science Australian National University ACT 2601, Australia bdm@cs.anu.edu.au (Received August 13, 2015) Abstract For each d > 0, we find all the smallest fullerenes for which the least distance between two pentagons is d. We also show that for each d there is an h d such that fullerenes with pentagons at least distance d apart and any number of hexagons greater than or equal to h d exist. We also determine the number of fullerenes where the minimum distance between any two pentagons is at least d, for 1 d 5, up to 400 vertices. 1 Introduction A fullerene [12] is a cubic plane graph where all faces are pentagons or hexagons. Euler s formula implies that a fullerene with n vertices contains exactly 12 pentagons and n/2 10 hexagons. The dual of a fullerene is the plane graph obtained by exchanging the roles of vertices and faces: the vertex set of the dual graph is the set of faces of the original graph and two vertices in the dual graph are adjacent if and only if the two faces share an edge in the original graph. The dual of a fullerene with n vertices is a triangulation (i.e. a plane graph where every face is a triangle) which contains 12 vertices with degree 5 and n/2 10 vertices with degree 6. The face-distance between two pentagons is the distance between the corresponding vertices of degree 5 in the dual graph. 1

2 The first fullerene molecule (i.e. the C 60 buckyball ) was discovered in 1985 by Kroto et al. [12]. Among the fullerenes, the Isolated Pentagon Rule (IPR) fullerenes are of special interest as they tend to be more stable [1, 16]. IPR fullerenes are fullerenes where no two pentagons share an edge, i.e. they have minimum face-distance at least 2. Raghavachari [13] argued that steric strain will be minimized when the pentagons are distributed as uniformly as possible and therefore proposed the uniform curvature rule as an extension of the IPR rule. Also, more recently Rodríguez-Fortea et al. [14] proposed the maximum pentagon separation rule where they argue that the most suitable carbon cages are those with the largest separations among the 12 pentagons. These observations lead us to investigate the maximum separation between pentagons that can be achieved for a given number of atoms, or conversely how many atoms are needed to achieve a given separation. We will refer to the least face-distance between pentagons of a fullerene as the pentagon separation of the fullerene. In the next section we determine the smallest fullerenes with a given pentagon separation. We also show that the minimum fullerenes for each d are unique up to mirror image and that for each d there is an h d such that fullerenes with pentagon separation at least d and any number of hexagons greater than or equal to h d exist. The latter was already proven for h 1 (i.e., for all fullerenes) by Grünbaum and Motzkin in [10] and for h 2 (i.e., for IPR fullerenes) by Klein and Liu in [11]. Finally, we also determine the number of fullerenes of pentagon separation d, for 1 d 5, up to 400 vertices. 2 Fullerenes with a given minimum pentagon separation In this section we determine the smallest fullerenes with a given pentagon separation. We remind the reader of the icosahedral fullerenes [5,9]. These fullerenes are uniquely determined by their Coxeter coordinates (p, q) and are obtained by cutting an equilateral Goldberg triangle with coordinates (p, q) from the hexagon lattice and gluing it to the faces of the icosahedron. As a Goldberg triangle with coordinates (p, q) has p 2 + pq + q 2 vertices, an icosahedral fullerene with Coxeter coordinates (p, q) has 20(p 2 + pq + q 2 ) vertices. Also note that an icosahedral fullerene with Coxeter coordinates (p, q) has pentagon separation p + q.

3 The smallest fullerene for d = 1 is of course unique: the icosahedron C 20. For larger d, the minimal fullerenes are given in the next theorem. Theorem 2.1. For odd d 3, the smallest fullerenes with pentagon separation at least d are the icosahedral fullerenes with Coxeter coordinates ( d/2, d/2 ) and ( d/2, d/2 ). These are mirror images and have 15d vertices. For even d, the unique smallest fullerene with pentagon separation at least d is the the icosahedral fullerene with Coxeter coordinates (d/2, d/2), which has 15d 2 vertices. Proof. Proof in the case that d 3 is odd: The penta-hexagonal net is the regular tiling of the plane where a central pentagon is surrounded by an infinite number of hexagons. The number of faces at face-distance k from the pentagon in the penta-hexagonal net is 5k. So the number of faces at face-distance at most k from the pentagon in the penta-hexagonal net is k 5i + 1 = 5k(k + 1)/ Figure 1 shows this situation for k = 3. i=1 Figure 1: Patch for d = 7 in the proof of Theorem 2.1 In a fullerene with pentagon separation at least d, for odd d, the sets of faces at face-distance at most d/2 from each pentagon are pairwise disjoint. Consequently the smallest such fullerenes we can hope to find consist of 12 copies of the above patch for k = d/2, which comes to 15d vertices. Since the patch boundary has no more than two consecutive vertices of degree 2, it is impossible to join any number of them into a larger patch with a boundary having more than two consecutive vertices of degree 2. Therefore, considering the complement, no union of these patches which is completable to a fullerene has more than two consecutive vertices of degree 3. Now, every way to overlap the boundaries of two patches produces

4 Figure 2: Bad and good ways to join two patches for d = 7 three consecutive vertices of degree 3, such as indicated in the left side of Figure 2, except for the way shown in the right side of Figure 2 or its mirror image. For each of these two starting points, there is only one way to attach a third patch to those two patches, and so on, leading to a unique completion in each case. It is easy to see that these two fullerenes are the icosahedral fullerenes mentioned in the theorem. Figure 3: Patch with dangling edges for d = 6 in the proof of Theorem 2.1 Proof in the case that d is even: The proof in this case is similar except that we use a different type of patch. In [6] it was proven that the number of vertices at distance k from the pentagon in the pentahexagonal net is 5 k/ So the total number of vertices at distance at most k from the pentagon in the penta-hexagonal net is k (5 i/2 + 5) = 5( k i/2 + k + 1). If k is even, i=0 k i/2 is equal to k 2 /4. So the total number of vertices at distance at most k from i=0 the pentagon in the penta-hexagonal net for even k is 5(k 2 /4 + k + 1). i=0

5 In a fullerene with pentagon separation at least d, for even d, the sets of vertices at distance at most d 2 from every pentagon are pairwise disjoint. The case of d = 6 is shown in Figure 3, excluding the ends of the dangling edges. Therefore, the smallest such fullerene of pentagon separation d we can hope to construct consists of 12 of these patches for k = d 2, joined together by identifying dangling edges. This would give us 15d 2 vertices altogether. Figure 4: Bad and good ways to identify dangling edges for d = 6 Since we are only permitted to create hexagons incident with the dangling edges, dangling edges distance two apart in one patch can only be identified with dangling edges distance two apart in another patch. Otherwise, a face of the wrong size is created, such as the pentagon indicated in the left side of Figure 4. This allows us to join two adjacent patches in only one way, as shown by the right side of Figure 4. Extra patches can then be attached in unique fashion, leading to a single fullerene that is easily seen to be the icosahedral fullerene with Coxeter coordinates (d/2, d/2). Next we will prove that for each d there is an h d such that fullerenes with pentagon separation at least d and any number of hexagons greater than or equal to h d exist. To prove this, we need Lemmas 2.2 and 2.3. A fullerene patch is a connected subgraph of a fullerene where all faces except one exterior face are also faces in the fullerene and all boundary vertices have degree 2 or 3 and all non-boundary vertices have degree 3. The boundary sequence of a patch is the cyclic sequence of the degrees of the vertices in the boundary of a patch in clockwise or counterclockwise order.

6 A cap is a fullerene patch which contains 6 pentagons and has a boundary sequence of the form (23) l (32) m. Such a boundary is represented by the parameters (l, m). In the literature, the vector (l, m) is also called the chiral vector (see [15]). Lemma 2.2. Any cap with parameters (l, 0) can be transformed into a cap with parameters (l, 1) without decreasing the minimum face-distance between the pentagons of the cap. Proof. Given a cap with parameters (l, 0). If the cap does not contain a pentagon in its boundary, we remove (l, 0) rings of hexagons until there is a pentagon in the boundary of the cap. In Figure 5 we show how the (l, 0) cap which contains a boundary pentagon (see Figure 5a) can be transformed into a cap with parameters (l, 1) without decreasing the minimum face-distance between the pentagons. This is done by changing the boundary pentagon into a hexagon h, adding a ring of hexagons (see Figure 5b) and changing a hexagon in the boundary which is adjacent to h into a pentagon (see Figure 5c). Lemma 2.3. Given a cap C with parameters (l, m) with l 0 and m 0 and which consists of f faces. A cap C with the same parameters (l, m) which contains C as a subgraph and has f + l, respectively f + m faces can be constructed from C by adding l or m hexagons to C, respectively. Proof. Given a cap C with parameters (l, m) with l 0 and m 0. In Figure 6 we show how a cap C with the same parameters (l, m) which contains C as a subgraph and has f + l faces can be constructed from C by adding l hexagons to C. A cap C with f + m faces can be obtained in a completely analogous way by adding m hexagons to C. Theorem 2.4. For each d there is an h d such that fullerenes with pentagon separation at least d and any number of hexagons greater than or equal to h d exist. Proof. Given an icosahedral fullerene F with Coxeter coordinates ( d/2, d/2 ). In this fullerene the minimum face-distance between the pentagons is 2 d/2. Brinkmann and Schein [4] have proven that every icosahedral fullerene with Coxeter coordinates (p, q) contains a fullerene patch with 6 pentagons which is a subgraph of a cap with parameters (3(p + 2q), 3(p q)). So F contains a fullerene patch with 6 pentagons which is a subgraph of a cap with parameters (9 d/2, 0).

7 (a) (b) (c) Figure 5: Procedure to change a cap with parameters (l, 0) to a cap with parameters (l, 1). The bold edges in the figure have to be identified with each other. It follows from [7, 15] that such a fullerene patch can be completed to a cap with parameters (9 d/2, 0) by adding hexagons. It follows from Lemma 2.2 that this cap can then be transformed to a cap with parameters (9 d/2, 1) without decreasing the minimum face-distance between the pentagons of the cap. We form a fullerene F with pentagon separation at least d by gluing together two copies of the (9 d/2, 1) cap and adding (9 d/2, 1) rings of hexagons if necessary. Let h F denote the number of hexagons of F. Now a fullerene with pentagon separation at least d and any number of hexagons greater than h F can be obtained by recursively applying Lemma 2.3 to F. The counts of the number of fullerenes up to 400 vertices with pentagon separation at least d, for 1 d 5, can be found in Tables 1-4. (Note that d = 1 gives the set of all fullerenes and d = 2 gives the set of all IPR fullerenes). These counts were

8 (a) (b) Figure 6: Procedure which adds l hexagons to an (l, m) cap without changing the boundary parameters. The bold edges in the figure have to be identified with each other. obtained by using the program buckygen [3, 8] (which can be downloaded from http: //caagt.ugent.be/buckygen/) to generate all non-isomorphic IPR fullerenes and then applying a separate program to compute their pentagon separation. Note that fullerenes which are mirror images of each other are considered to be in the same isomorphism class and are thus only counted once. Some of the fullerenes from Tables 1-4 can also be downloaded from the House of Graphs [2] at Figures 7-9 show the smallest fullerenes with pentagon separation d, for 3 d 5. Acknowledgements: Jan Goedgebeur is supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO). Brendan McKay is supported by the Australian Research Council. Most computations for this work were carried out using the Stevin Supercomputer Infrastructure at Ghent University. We also would like to thank Gunnar Brinkmann, Patrick Fowler and Jack Graver for useful suggestions. References [1] E. Albertazzi, C. Domene, P.W. Fowler, T. Heine, G. Seifert, C. Van Alsenoy, and F. Zerbetto. Pentagon adjacency as a determinant of fullerene stability. Physical Chemistry Chemical Physics, 1(12): , 1999.

9 nv nf fullerenes IPR fullerenes pent. sep. 3 pent. sep. 4 pent. sep Table 1: Number of fullerenes for a given lower bound on the pentagon separation. nv is the number of vertices and nf is the number of faces.

10 nv nf fullerenes IPR fullerenes pent. sep. 3 pent. sep. 4 pent. sep Table 2: Number of fullerenes for a given lower bound on the pentagon separation (continued). nv is the number of vertices and nf is the number of faces.

11 nv nf fullerenes IPR fullerenes pent. sep. 3 pent. sep. 4 pent. sep Table 3: Number of fullerenes for a given lower bound on the pentagon separation (continued). nv is the number of vertices and nf is the number of faces.

12 nv nf fullerenes IPR fullerenes pent. sep. 3 pent. sep. 4 pent. sep Table 4: Number of fullerenes for a given lower bound on the pentagon separation (continued). nv is the number of vertices and nf is the number of faces.

13 Figure 7: The icosahedral fullerene with Coxeter coordinates (2, 1). This fullerene and its mirror image are the smallest fullerenes with pentagon separation 3. They have 140 vertices. Figure 8: The icosahedral fullerene with Coxeter coordinates (2, 2). fullerene with pentagon separation 4 and has 240 vertices. This is the smallest [2] G. Brinkmann, K. Coolsaet, J. Goedgebeur, and H. Mélot. House of Graphs: a database of interesting graphs. Discrete Applied Mathematics, 161(1-2): , Available at [3] G. Brinkmann, J. Goedgebeur, and B.D. McKay. The generation of fullerenes. Journal of Chemical Information and Modeling, 52(11): , [4] G. Brinkmann and S. Schein. Personal communication.

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15 [13] K. Raghavachari. Ground state of C 84 : two almost isoenergetic isomers. Chemical Physics Letters, 190(5): , [14] A. Rodríguez-Fortea, N. Alegret, A.L. Balch, and J.M. Poblet. The maximum pentagon separation rule provides a guideline for the structures of endohedral metallofullerenes. Nature Chemistry, 2(11): , [15] R. Saito, G. Dresselhaus, and M.S. Dresselhaus. Physical properties of carbon nanotubes, volume 4. World Scientific, [16] T.G. Schmalz, W.A. Seitz, D.J. Klein, and G.E. Hite. Elemental carbon cages. Journal of the American Chemical Society, 110(4): , 1988.